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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An elementary proof of Titchmarsh's convolution theorem


Author: Raouf Doss
Journal: Proc. Amer. Math. Soc. 104 (1988), 181-184
MSC: Primary 42A85; Secondary 45E10
MathSciNet review: 958063
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Abstract: We give an elementary proof of the following theorem of Titchmarsh. Suppose $ f,g$ are integrable on the interval $ \left( {0,2T} \right)$ and that the convolution $ f * g\left( t \right) = \int_0^t {f\left( {t - x} \right)g\left( x \right)dx} = 0$ on $ \left( {0,2T} \right)$. Then there are nonnegative numbers $ \alpha ,\beta $ with $ \alpha + \beta \geq 2T$ for which $ f\left( x \right) = 0$ for almost all $ x$ in $ \left( {0,\alpha } \right)$ and $ g\left( x \right) = 0$ for almost all $ x$ in $ \left( {0,\beta } \right)$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0958063-5
PII: S 0002-9939(1988)0958063-5
Article copyright: © Copyright 1988 American Mathematical Society